arXiv 2401.047842024-01quant-ph

量子近似优化算法的能量景观

Energy Landscapes for the Quantum Approximate Optimisation Algorithm

Boy Choy, David J. Wales

AI总结：

本文利用盆地跳跃与离散路径采样分析QAOA能量景观，发现其单漏斗结构及次低极小值可能更优，并提出基于期望阈值的性能评估指标。

摘要：

变分量子算法（VQAs）在当代含噪声中等规模量子（NISQ）时代，已展现出解决NP难组合问题的巨大潜力。量子近似优化算法（QAOA）正是此类算法之一，它通过在相应的Trotter化拟设中连续执行$L$层量子电路，用于求解给定图的最大割（Max-Cut）问题。随着电路深度增加，局部极小值呈指数级激增，由此带来的VQA代价函数探索挑战已有充分记录。然而，研究电路深度对QAOA找到正确Max-Cut解的性能影响的工作较少。本文采用盆地跳跃全局优化方法，在多种图的QAOA拟设能量景观中导航，并分析QAOA找到正确Max-Cut解的性能。我们还利用离散路径采样研究解空间结构，构建局部极小值及连接它们的过渡态数据库，并通过断连图提供富有洞察力的可视化。结果发现，相应景观通常具有单一漏斗组织，这使得以良好的Max-Cut解概率定位低位极小值相对直接。在绝热极限以下的某些情况下，第二低的局部极小值甚至可能产生比全局极小值更高的解概率。这一重要观察促使我们基于盆地跳跃全局优化获得的极小值集合，开发评估QAOA性能的更广泛指标。因此，我们建立了期望阈值，用于阐明来自局部极小值的有用解概率，该方法可能在阐明局部极小值的合理解概率方面带来显著收益。

英文摘要：

Variational quantum algorithms (VQAs) have demonstrated considerable potential in solving NP-hard combinatorial problems in the contemporary near intermediate-scale quantum (NISQ) era. The quantum approximate optimisation algorithm (QAOA) is one such algorithm, used in solving the maximum cut (Max-Cut) problem for a given graph by successive implementation of $L$ quantum circuit layers within a corresponding Trotterised ansatz. The challenge of exploring the cost function of VQAs arising from an exponential proliferation of local minima with increasing circuit depth has been well-documented. However, fewer studies have investigated the impact of circuit depth on QAOA performance in finding the correct Max-Cut solution. Here, we employ basin-hopping global optimisation methods to navigate the energy landscapes for QAOA ansätze for various graphs, and analyse QAOA performance in finding the correct Max-Cut solution. The structure of the solution space is also investigated using discrete path sampling to build databases of local minima and the transition states that connect them, providing insightful visualisations using disconnectivity graphs. We find that the corresponding landscapes generally have a single funnel organisation, which makes it relatively straightforward to locate low-lying minima with good Max-Cut solution probabilities. In some cases below the adiabatic limit the second lowest local minimum may even yield a higher solution probability than the global minimum. This important observation has motivated us to develop broader metrics in evaluating QAOA performance, based on collections of minima obtained from basin-hopping global optimisation. Hence we establish expectation thresholds in elucidating useful solution probabilities from local minima, an approach that may provide significant gains in elucidating reasonable solution probabilities from local minima.

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